inflection point and invervals

Question

Determine the intervals on which the given function is concave up or down and find the points of inflection. Let f(x)=(x^2-8)e^x

Answer 1

f'(x)'=(x²-8)eˣ+2xeˣ=eˣ(x²+2x-8)=eˣ(x-2)(x+4). When f'(x)=0 at a change of direction of the gradient (extremum) x=2 and -4. f(2)=-4e² and f(-4)=8/e⁴. For x between −∞ and -4, f(x) is increasing (concave up) and between -4 and 2 f(x) is decreasing (concave down) to the minimum, after which f(x) is increasing (concave up).